Harish-Chandra induction and restriction for finite general linear groups

Levi decompositions. Let $G$ be a finite group and let $P\le G$ be a subgroup that is the internal semidirect product $P=L⋉U$ of a subgroup $L\le P$ and a normal subgroup $U⊴P$ with $P=LU$ and $L\cap U=\{1\}$ (Subgroup, Normal subgroup: invariance under conjugation); one calls $P$ a parabolic subgroup with Levi complement $L$ and unipotent radical $U$ when $P$ arises this way from the finite general linear group below. The projection $\pi :P⟶L,\pi (lu):=l(l\in L,u\in U),$ is a surjective group homomorphism with kernel $U$: it is the composite of the quotient map $P\to P\mathrm{/}U$ of The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$ with the inverse of the isomorphism $L\to P\mathrm{/}U$, $l\mapsto lU$, which is bijective because $P=LU$ and $L\cap U=\{1\}$ (First isomorphism theorem for groups: $G\mathrm{/}\ker f\cong \mathrm{im}f$); hence $P\mathrm{/}U\cong L$.

Inflation. Let $R$ be a commutative ring and let $V$ be an $R$-linear $L$-module (An $R$-linear action of $G$ on a left $R$-module, and a $G$-module over $R$). The inflation of $V$ from $L$ to $P$ is the $R$-module $V$ together with the $P$-action $p\cdot v:=\pi (p)\cdot v(p\in P,v\in V),$ which is well defined because $\pi$ is a homomorphism, and under which every element of $U$ acts as the identity; we write ${\mathrm{Inf}}_L^{P}V$ for this $R$-linear $P$-module.

Harish-Chandra induction. With $V$ as above, the Harish-Chandra induction of $V$ from $L$ to $G$ (with respect to the parabolic $P$) is the $R$-linear $G$-module $R_L^G(V):={\mathrm{Ind}}_P^G\text{\hspace{0.17em}⁣}\left({\mathrm{Inf}}_L^{P}V\right),$ the induction from $P$ to $G$ of The induced $R$-linear $G$-module ${\mathrm{Ind}}_H^{G}W$ as $H$-covariant functions on $G$; thus $R_L^G(V)$ is the set of functions $f:G\to V$ with $f(gp)=\pi (p{)}^{-1}f(g)$ for all $g\in G$, $p\in P$, and $(x\cdot f)(g)=f(x^{-1}g)$.

Harish-Chandra restriction. Let $X$ be an $R$-linear $G$-module. Its $U$-invariants are $X^U:=\{x\in X:u\cdot x=x\text{ for every }u\in U\},$ an $R$-submodule of $X$ that is stable under $P$. Define the Harish-Chandra restriction of $X$ from $G$ to $L$ (with respect to $P$) to be $X^U$ equipped with the $L$-action $l\cdot x:=lx(l\in L,x\in X^U),$ the restriction of the $P$-action. This is well defined as an action of $L$, and it depends only on the class of $l$ in $P\mathrm{/}U\cong L$: if $p\in P$ has $\pi (p)=l$, then $p=lu$ with $u\in U$ and $px=lux=lx$ for $x\in X^U$. We write ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^G(X):=X^U$ for this $R$-linear $L$-module.

Functoriality. If $\phi :V\to V^{\mathrm{\prime }}$ is a morphism of $R$-linear $L$-modules, then $\phi$ is also $P$-linear for the inflated actions, and postcomposition with $\phi$ defines a morphism $R_L^G(\phi ):R_L^G(V)\to R_L^G(V^{\mathrm{\prime }})$ of $R$-linear $G$-modules, because $(\phi \circ f)(gp)=\pi (p{)}^{-1}\phi (f(g))$; the assignments $V\mapsto R_L^G(V)$ and $X\mapsto {}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^G(X)$ are additive functors between the categories of $R$-linear $L$-modules and $R$-linear $G$-modules, and $X\mapsto X^U$ acts on morphisms by restriction since the action of $U$ is $R$-linear by An $R$-linear action of $G$ on a left $R$-module, and a $G$-module over $R$.

The case of finite general linear groups. Now let $n\ge 1$, let $q$ be a prime power and put $G={\mathrm{GL}}_n({\mathbb{F}}_q)$, with diagonal torus $T$ and standard flag $V_{\bullet }$ (Standard subgroups of finite general linear groups). For a composition $\alpha$ of $n$ the standard parabolic subgroup $P_{\alpha }$ has the Levi decomposition $P_{\alpha }=L_{\alpha }⋉U_{\alpha }$ (Compositions, partial flags, and standard parabolics, Block Levi decomposition of standard parabolics), so the constructions above apply with $P=P_{\alpha }$, $L=L_{\alpha }$, $U=U_{\alpha }$; we write $R_{L_{\alpha }}^G(V):={\mathrm{Ind}}_{P_{\alpha }}^G\text{\hspace{0.17em}⁣}\left({\mathrm{Inf}}_{L_{\alpha }}^{P_{\alpha }}V\right),{}^{\ast }\text{\hspace{0.17em}⁣}{R}_{L_{\alpha }}^G(X):=X^{U_{\alpha }}\mathrm{.}$ More generally, a subgroup of $G$ is a split Levi subgroup when it is of the form $g{L}_{\alpha }{g}^{-1}$ for some $g\in G$ and composition $\alpha$, and a split parabolic subgroup is a subgroup of the form $g{P}_{\alpha }{g}^{-1}$; such a subgroup has the Levi decomposition $g{P}_{\alpha }{g}^{-1}=(g{L}_{\alpha }{g}^{-1})⋉(g{U}_{\alpha }{g}^{-1})$, and the general construction above attaches to it the functors $R_L^G$ and ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^G$ for $L=g{L}_{\alpha }{g}^{-1}$. For an arbitrary split parabolic the definitions therefore involve no further choice: the parabolic subgroup $P$ and its unipotent radical $U$ are part of the data, and the notation $R_L^G$, ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^G$ records only the Levi. For complex modules and coordinate parabolics obtained by ordering the same coordinate blocks, the parabolic-independence theorem of this page proves that these functors depend, up to natural isomorphism, only on $L$. Over a general coefficient ring $R$, the parabolic $P$ remains part of the functor data; when needed we display it as $R_L^{G,P}$ and ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^{G,P}$.

Standing conventions. Except where another coefficient ring is named, all modules on this page are complex vector spaces, so $R=\mathbb{C}$, the group algebras are semisimple by Maschke's theorem for finite groups over fields whose characteristic does not divide $\mathrm{\mid }G\mathrm{\mid }$, and the functors above are the complex Harish-Chandra functors of Dudas–Michel. For finite-dimensional $V$ one has ${\dim }_{\mathbb{C}}{R}_L^G(V)=[G:P]{\dim }_{\mathbb{C}}V$, because an induced module is free over the index of $P$ in $G$ (the functions of The induced $R$-linear $G$-module ${\mathrm{Ind}}_H^{G}W$ as $H$-covariant functions on $G$ are determined by their values on a set of left coset representatives), and ${\dim }_{\mathbb{C}}{}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^G(X)\le {\dim }_{\mathbb{C}}X$.